A microsphere-homogenized strain gradient elasticity model for polymers

Polymers consist of many discrete chains, making them inherently discrete rather than continuous. To analyze polymers (and their composites) using continuum mechanics, it is necessary to establish a bridge between their discrete and continuum models. In this paper, the discrete microsphere model is employed to derive a physics-based strain gradient continuum, where the strain gradient term relies on the concrete geometric structure. This is achieved by connecting the stretch fluctuation field of polymer chains with the strain gradient field through an asymptotic homogenization method. This homogenization method first provides the construction of the Helmholtz free energy density for the microsphere model and then develops the transformation of the free energy density to that strain gradient continuum. Applying the proposed strain gradient continuum to the Euler–Bernoulli beam, the size-dependent effects of the free energy, the bending rigidity, and deflection are investigated in detail. This homogenization method bridges the gap between discrete and continuous polymer mediums. Furthermore, the continuum model retains high-order strain gradient information. This correlation facilitates the application of polymers in nanocomposites, enabling the creation of groundbreaking materials through artificial design.